Nonlinear Autonomous Dynamical Systems Research Articles

Cascade tri-neuron hopfield neural network: Dynamical analysis and analog circuit implementation

This paper studies a cascade tri-neuron Hopfield neural network (CTN-HNN) with no connection between the first neuron and the third neuron. Such incompletely connected neuronal structure may be part of complex neural networks with negligible inputs, and may be present in abnormal neural networks with some neurological diseases. It was proved that the stable point, limit cycle, single-scroll chaotic attractor and double-scroll chaotic attractor were observed in the network by adjusting the synaptic weight from the second neuron to itself. To further dynamic analysis, we first demonstrate that this three-dimensional (3D) autonomous nonlinear dynamical system is symmetric about the origin and bounded ultimately, and has one unstable saddle-focus zero point and two symmetric nonzero points with different stability types. Afterwards, taking two synaptic weights as adjustable parameters, the numerical analysis reveals that the CTN-HNN exhibits coexisting chaotic and periodic attractors, coexisting bursting firings and long-term transient chaos by employing multiple numerical methods. Finally, an analog printed circuit is built using one main circuit module and three activation function modules, thus effectively demonstrating the complex dynamics of the network.

Model Representation of Adjustable Differentiation of Regional Socio-Economic Systems

Introduction. According to the law of differentiation, the uneven development of the elements of the system leads to its heterogeneity and is a necessary precondition for the process of evolution. By weakening the links between the elements of the system, differentiation initiates disturbances during coordination processes. The objectively existing contradiction in the organization of socio-economic systems between coordination and differentiation can be overcome by managing the process of differentiation of the elements of the system. The aim of the study is to build an economic and mathematical model of regulated differentiation of regional socio-economic systems (subjects of the Russian Federation, the country as a whole), allowing to establish the conditions and limitations of equilibrium trajectories of their functioning, as well as to determine the parameters of differentiation that stimulates economic growth. Materials and Methods. The hypothesis of the study is an assumption about the influence of intra-territorial differentiation on economic growth and socio-economic development of the region. The methodological approach based on the representation of the model of uneven development of territories in comparison with the dynamic characteristics of their economic development is proposed and implemented. The approach is based on the study of the properties of nonlinear autonomous dynamic systems. The model is presented in the form of a system of differential equations in the space of values of the decile coefficient of differentiation and gross regional product per capita. Results. The calculations carried out on statistical data for the constituent entities of the Russian Federation for 2 years empirically confirmed the hypothesis put forward about the impact of intra-territorial differentiation on the economic development of the constituent entities of the Russian Federation. The results obtained complement the theoretical ideas about the stimulating effects of uneven territorial development and confirm the consistency of the provisions of the concept of regulated differentiation of socio-economic development of territories. Discussion and Conclusion. The scope of application of the research results consists in the possibility of building a dynamic economic-mathematical model of regulated differentiation of regional socio-economic systems (subjects of the Russian Federation, the country as a whole), allowing to determine the parameters of differentiation that stimulates economic growth, and the development on its basis of normative strategies for territorial development.

Introducing some classes of stable systems without any smooth Lyapunov functions

This paper proves that certain classes of stable nonlinear systems do not admit any smooth Lyapunov functions. In fact, the stability analysis of two different classes of nonlinear dynamical systems is provided, and it is demonstrated that their stability properties cannot be established by any convex Lyapunov functions or smooth Lyapunov functions. In this regard, we begin by studying our first class of autonomous dynamical systems and prove that, despite stability, there are no convex Lyapunov functions in the form of V(x) or V(t,x) to establish the stability properties. For the second class, we consider a much more general form of nonlinear autonomous dynamical systems with a stable origin, and prove that this class, too, does not admit any smooth Lyapunov functions in the form of V(x) or V(t,x). Furthermore, another general class of stable non-autonomous dynamical systems is presented, and it is demonstrated that there is no smooth Lyapunov function in the form of V(x) to establish the stability properties. Finally, for the second class of more general stable non-autonomous systems, it is proved that the class does not admit even a continuous Lyapunov function in the form of V(x) or V(t,x).

Counting equilibria in a random non-gradient dynamics with heterogeneous relaxation rates

We consider a nonlinear autonomous random dynamical system of N degrees of freedom coupled by Gaussian random interactions and characterized by a continuous spectrum n μ (λ) of real positive relaxation rates. Using Kac–Rice formalism, the computation of annealed complexities (both of stable equilibria and of all types of equilibria) is reduced to evaluating the averages involving the modulus of the determinant of the random Jacobian matrix. In the limit of large system N ≫ 1 we derive exact analytical results for the complexities for short-range correlated coupling fields, extending results previously obtained for the ‘homogeneous’ relaxation spectrum characterised by a single relaxation rate. We show the emergence of a ‘topology trivialisation’ transition from a complex phase with exponentially many equilibria to a simple phase with a single equilibrium as the magnitude of the random field is decreased. Within the complex phase the complexity of stable equilibria undergoes an additional transition from a phase with exponentially small probability to find a single stable equilibrium to a phase with exponentially many stable equilibria as the fraction of gradient component of the field is increased. The behaviour of the complexity at the transition is found only to depend on the small λ behaviour of the spectrum of relaxation rates n μ (λ) and thus conjectured to be universal. We also provide some insights into a counting problem motivated by a paper of Spivak and Zyuzin of 2004 about wave scattering in a disordered nonlinear medium.

Identification of autonomous nonlinear dynamical system based on discrete-time multiscale wavelet neural network

Identification of autonomous nonlinear dynamical system based on discrete-time multiscale wavelet neural network

Comments on “Stability Regions of Nonlinear Autonomous Dynamical Systems”

The proofs of the groundbreaking theorems of [Chiang, Hirsch, and Wu. Stability Regions of Nonlinear Autonomous Dynamical Systems, IEEE Transactions on Automatic Control, 1988] rely on a lemma which states that if the stable manifold of a first hyperbolic closed orbit intersects transversely the unstable manifold of a second (possibly the same) hyperbolic closed orbit, then the dimension of the unstable manifold of the first is strictly less than the dimension of the unstable manifold of the second. However, we provide an example meeting the conditions of the lemma where the dimensions of the unstable manifolds are equal, thereby disproving the lemma. In particular, we present a hyperbolic closed orbit of a smooth vector field over three-dimensional Euclidean space whose stable and unstable manifolds have nonempty, transverse intersection.

Koopman Resolvent: A Laplace-Domain Analysis of Nonlinear Autonomous Dynamical Systems

The motivation of our research is to establish a Laplace-domain theory that provides principles and methodology to analyze and synthesize systems with nonlinear dynamics. A semigroup of composition operators defined for nonlinear autonomous dynamical systems -- the Koopman semigroup and its associated Koopman generator -- plays a central role in this study. We introduce the resolvent of the Koopman generator, which we call the Koopman resolvent, and provide its spectral characterization for three types of nonlinear dynamics: ergodic evolution on an attractor, convergence to a stable equilibrium point, and convergence to a (quasi-)stable limit cycle. This shows that the Koopman resolvent provides the Laplace-domain representation of such nonlinear autonomous dynamics. A computational aspect of the Laplace-domain representation is also discussed with emphasis on non-stationary Koopman modes.

On the dynamics of a modified Lorenz–Stenflo system

The Lorenz–Stenflo system is a four-parameter four-dimensional autonomous nonlinear continuous-time dynamical system, derived to model the time evolution of finite amplitude acoustic gravity waves in a rotating atmosphere. In this paper, we propose a modified Lorenz–Stenflo system, where the variable [Formula: see text] in the fourth equation of the original Lorenz–Stenflo system was replaced by [Formula: see text]. We investigate cross-sections of the parameter-space of this new system, characterizing regions of different dynamical behaviors. We show that the aforementioned replacement may promote the emergence of organized periodic structures in places of these cross-sections, where they did not exist before modification.

Signal flow graph and control of realizable autonomous nonlinear Chen model with quaternion variables

In this work, we continue our study of autonomous nonlinear dynamical systems with quaternion variables. We present a novel chaotic Chen system with quaternion variables. This system in real form is high (nine) dimensional. We study the dynamics and basic properties of this model with quaternion variables. We also evaluate the stability of the trivial points and restrict the conditions under which the nonlinear novel quaternion Chen system has negative, zero, or positive Lyapunov exponents. A signal flow graph and an electronic circuit implementation are designed, analyzed, and constructed to realize the novel Chen system. The control problem of chaotic nonlinear systems with quaternion variables is investigated. We propose and design an approach to build a controller for these systems. The viability and usefulness of the suggested approach are illustrated by an application to the chaotic Chen model with quaternion variables. Using the proposed method, the attractors of the quaternion Chen model are transformed to trivial fixed point, quasi-periodic, or periodic (limit cycle) states. The mathematical results for the controller according to this method are confirmed numerically.

Minimal blowing pressure allowing periodic oscillations in a simplified reed musical instrument model: Bouasse-Benade prescription assessed through numerical continuation

A reed instrument model with N acoustical modes can be described as a 2N dimensional autonomous nonlinear dynamical system. Here, a simplified model of a reed-like instrument having two quasi-harmonic resonances, represented by a four dimensional dynamical system, is studied using the continuation and bifurcation software AUTO. Bifurcation diagrams of equilibria and periodic solutions are explored with respect to the blowing mouth pressure, with focus on amplitude and frequency evolutions along the different solution branches. Equilibria and periodic regimes are connected through Hopf bifurcations, which are found to be direct or inverse depending on the physical parameters values. Emerging periodic regimes mainly supported by either the first acoustic resonance (first register) or the second acoustic resonance (second register) are successfully identified by the model. An additional periodic branch is also found to emerge from the branch of the second register through a period-doubling bifurcation. The evolution of the oscillation frequency along each branch of the periodic regimes is also predicted by the continuation method. Stability along each branch is computed as well. Some of the results are interpreted in terms of the ease of playing of the reed instrument. The effect of the inharmonicity between the first two impedance peaks is observed both when the amplitude of the first is greater than the second, as well as the inverse case. In both cases, the blowing pressure that results in periodic oscillations has a lowest value when the two resonances are harmonic, a theoretical illustration of the Bouasse-Benade prescription.

Dynamical system analysis of Hessence scalar field in teleparallel gravity: Invariant manifold technique

This paper investigates the cosmological dynamics of the Hessence scalar field coupled with the dark matter in the background of the teleparallel gravity. We have assumed that the potential of the scalar field is exponential in nature whereas the [Formula: see text] appearing in the teleparallel theory has the form [Formula: see text]. The field equations of this system reduce to a nonlinear autonomous system and dynamical system analysis is then performed. Due to the nonlinearity and the existence of multiple zero eigenvalues, the traditional procedures of analysis break down. So some novel technique is required. One of the latest such techniques is the invariant manifold theory. By the application of this theory, one projects the variables linked with the zero eigenvalues onto the variables linked with the nonzero eigenvalues to compute the center manifolds and the reduced systems associated with the critical points. These reduced systems reflect the nature of the whole dynamical systems. They also have less dimension and are often simple in nature. Hence, it is possible to solve them directly. In this paper, we work exactly in this spirit and find the center manifolds and solve the corresponding reduced system for some of the critical points associated with the dynamical system. We discover some interesting results namely that there are certain bounds on the interaction term [Formula: see text] which asserts the stability of the systems. We also present various stability diagrams of the reduced systems. An asymptotic analysis is then done for the critical points at infinity. Finally, we discuss the cosmological interpretation of our results.

Stability of Transactive Energy Market-Based Power Distribution System Under Data Integrity Attack

Future smart power distribution system with multiple active consumers and aggregators exchanging real-time electricity pricing and electricity consumption information over a communication network is prone to cyberattack. The goal of this paper is to understand the impact of attacks on pricing/load signals on the physical grid within a transactive energy market framework. First, this paper models the interaction between real-time electricity price and total energy demand in the form of a discrete time nonlinear autonomous dynamical system. Second, equilibrium electricity price and energy demand associated with this coupled dynamical system is derived and conditions for bounded input bounded output (BIBO) stability are identified. Third, a BIBO stable algorithm to design real-time electricity pricing scheme from a techno-economic perspective is developed. Finally, the impact of various levels of false data injection (FDI) attack on price of electricity, demand, and distribution system voltage is investigated. The proposed model is analyzed using simulations on the IEEE 69-bus test system and the impact of FDI attack on both electricity price/demand and distribution grid voltage is quantified. This paper shows that impact of FDI attack on electricity prices is more severe than an attack on electricity demand.

Integrable cosmological model with van der Waals gas and matter creation

A cosmological model with van der Waals gas and dust has been studied in the context of a three-component autonomous non-linear dynamical system involving the time evolution of the particle number density, the Hubble parameter and the temperature. Due to the presence of a symmetry of the model, the temperature evolution law is determined (in terms of the particle number density) and with this the dynamical system reduces to a two-component one which is fully integrable. The globally conserved Hamiltonian is identified and, in addition to it, some special (second) integrals, defined and conserved on a lower-dimensional manifold, are found. The parameter choices and their implication for the global dynamics in terms of cosmological relevance are comprehensively studied and the physically meaningful parameter values are identified.

Van der Waals universe with adiabatic matter creation

A FRWL cosmological model with perfect fluid comprising of van der Waals gas and dust has been studied in the context of dynamical analysis of a three-component autonomous non-linear dynamical system for the particle number density n, the Hubble parameter H, and the temperature T. Perfect fluid isentropic particle creation at rate proportional to an integer power alpha of H has been incorporated. The existence of a global first integral allows the determination of the temperature evolution law and hence the reduction of the dynamical system to a two-component one. Special attention is paid to the cases of alpha = 2 and alpha = 4 and these are illustrated with numerical examples. The global dynamics is comprehensively studied for different choices of the values of the physical parameters of the model. Trajectories in the (n, H) phase space are identified for which temporary inflationary regime exists.

Efficient spectral domain technique for the frequency locking analysis of nonlinear oscillators

After discussing an implementation of the harmonic balance technique that enables the efficient determination of the limit cycles for a nonlinear autonomous dynamical system, we consider the frequency locking of a set of oscillators that is studied by means of a proper extension of the aforementioned approach. Harmonic balance is also used for the numerical computation of the Floquet exponents and eigenvectors of the frequency locked limit cycle, thus enabling the assessment of its stability properties. The proposed technique is applied to the study of the frequency locking properties of a set of coupled Chua’s oscillators as a function of several parameters.

Learning position and orientation dynamics from demonstrations via contraction analysis

This paper presents a unified framework of model-learning algorithms, called contracting dynamical system primitives (CDSP), that can be used to learn pose (i.e., position and orientation) dynamics of point-to-point motions from demonstrations. The position and the orientation (represented using quaternions) trajectories are modeled as two separate autonomous nonlinear dynamical systems. The special constraints of the $${\mathbb {S}}^{3}$$ manifold are enforced in the formulation of the system that models the orientation dynamics. To capture the variability in the demonstrations, the dynamical systems are estimated using Gaussian mixture models (GMMs). The parameters of the GMMs are learned subject to the constraints derived using partial contraction analysis. The learned models’ reproductions are shown to accurately reproduce the demonstrations and are guaranteed to converge to the desired goal location. Experimental results illustrate the CDSP algorithm’s ability to accurately learn position and orientation dynamics and the utility of the learned models in path generation for a Baxter robot arm. The CDSP algorithm is evaluated on a publicly available dataset and a synthetic dataset, and is shown to have the lowest and comparable average reproduction errors when compared to state-of-the-art imitation learning algorithms.

A three-dimensional autonomous nonlinear dynamical system modelling equatorial ocean flows

We investigate a nonlinear three-dimensional model for equatorial flows, finding exact solutions that capture the most relevant geophysical features: depth-dependent currents, poleward or equatorial surface drift and a vertical mixture of upward and downward motions.

Saddle-node equilibrium points on the stability boundary of nonlinear autonomous dynamical systems

ABSTRACTA complete characterization of the stability boundary of an asymptotically stable equilibrium point in the presence of type-k saddle-node non-hyperbolic equilibrium points, with k ≥ 0, on the stability boundary is developed in this paper. Under the transversality condition, it is shown that the stability boundary is composed of the stable manifolds of the hyperbolic equilibrium points on the stability boundary, the stable manifolds of type-0 saddle-node equilibrium points on the stability boundary and the stable centre and centre manifolds of the type-r saddle-node equilibrium points with r ≥ 1 on the stability boundary. This characterization is the first step to understanding the behaviour of stability regions and stability boundaries in the occurrence of saddle-node bifurcations on the stability boundary.

Dynamics of switching van der Pol circuits

This paper addresses dynamical behaviors of switching van der Pol circuits by investigating stability issues of autonomous nonlinear dynamical systems. Firstly, a kind of nonlinear dynamical systems, i.e., a piecewise/ switching autonomous nonlinear system, is used to formulate a class of van der Pol circuits that are composed of at least two alternative electrical sub-circuits. Secondly, based on the concept of system stability in the sense of Lyapunov and continuously positive definite functions, this paper proposes two new necessary and sufficient conditions of stability/ asymptotic stability for autonomous nonlinear systems. Thirdly, based on the new stability results obtained in this paper, we present several criteria for globally asymptotical stability and local instability of the unique zero equilibrium of such kind of electrical circuits. Meanwhile, the existence and uniqueness of a stable limit cycle are also given for the switching van der Pol circuit. These novel results obtained in this paper show that switching actions involved in van der Pol circuits may vanish the classical nonlinear dynamical phenomena of relaxation oscillations. Finally, numerical simulations of several nonlinear systems and a switching van der Pol circuit illustrate the effectiveness and practicality of our new results.

The Gesture as an Autonomous Nonlinear Dynamical System

ABSTRACTWe propose a theory of how the speech gesture determines change in a functionally relevant variable of vocal tract state (e.g., constriction degree). A core postulate of the theory is that the gesture determines how the variable evolves in time independent of any executive timekeeper. That is, the theory involves intrinsic timing of speech gestures. We compare the theory against others in which an executive timekeeper determines change in vocal tract state. Theories that employ an executive timekeeper have been proposed to correct for disparities between theoretically predicted and experimentally observed velocity profiles. Such theories of extrinsic timing make the gesture a nonautonomous dynamical system. For a nonautonomous dynamical system, the change in state depends not just on the state but also on time. We show that this nonautonomous extension makes surprisingly weak kinematic predictions both qualitatively and quantitatively. We propose instead that the gesture is a theoretically simpler nonlinear autonomous dynamical system. For the proposed nonlinear autonomous dynamical system, the change in state depends nonlinearly on the state and does not depend on time. This new theory provides formal expression to the notion of intrinsic timing. Furthermore, it predicts experimentally observed relations among kinematic variables.